The Pythagorean Triple Essay Example for Free

The Pythagorean Triple Essay Pythagoras a Greek philosopher and mathematician is very famous for its Pythagorean Theorem. This theorem states that if a, b and c are sides of a right triangle then a2 + b2 = c2 (Morris, 1997). The study of the Pythagorean triples started long before Pythagoras knew how to solve it. There were evidences that Babylonians have lists of the triples written in a tablet. This would only mean that Babylonians may have known a method on how to produce such triples (Silverman, 16 – 26). Pythagorean triple is a set of number consisting of three natural numbers that can suit the Pythagorean equation a2 + b2 = c2. Some of the known triples are 3, 4, 5 and 5, 12, 13 (Bogomolny, 1996). How can we derive such triples? If we multiple the Pythagorean formula by 2 then we generate another formula 2a2 + 2b2 = 2c2. This only means that if we multiply 2 to the Pythagorean triple 3, 4, 5 and 5, 12, 13 then we can get another set of Pythagorean triple. The answer to that is triple 6, 8, 10 and 10, 24, 26. To check whether the said triple are Pythagorean triple, we can substitute it to the original formula a2 + b2 = c2. Check: is 6, 8, 10 Pythagorean triple? 62 + 82 = 102 36 + 64 = 100 100 = 100 Thus 6, 8 and 10 satisfy the Pythagorean equation. Δ 6, 8, 10 is a Pythagorean triple. Check: is 10, 24, 26 satisfy the Pythagorean equation? 102 + 242 = 262 100 + 576 = 676 676 = 676 Thus 10, 24, 26 satisfy the Pythagorean equation. Δ 10, 24, 26 is a Pythagorean triple. If we multiply the Pythagorean equation by 3 and using the first 2 Pythagorean triple mentioned above, we can yield another set of Pythagorean triple. Thus we can formulate a general formula that can produce different sets of Pythagorean triple. We can generate an infinite number of Pythagorean triple by using the Pythagorean triple 3, 4, 5. If we multiple d, where k is an integer, to that triple we will yield different sets of Pythagorean triple all the time. d*(3, 4, 5) where d is an integer. Check: if k is equal to 4 we get a triple 12, 16, and 20. Is this a Pythagorean triple? By substitution, 122 + 162 = 202 144 + 256 = 400 400 = 400 Thus 12, 16, 20 satisfy the Pythagorean equation. Δ 12, 16, 20 is a Pythagorean triple. Check: if k is equal to 5 we get a triple 15, 20, 25. Is this a Pythagorean triple? By substitution, 152 + 202 = 252 225 + 400 = 625 625 = 625 Thus 15, 20, 25 satisfy the Pythagorean equation. Δ 15, 20, 25 is a Pythagorean triple. But the formula given above is just a formula for getting the multiples of the Pythagorean triple. But is there a general formula in getting these triples? There are formulas that can solve each and every Pythagorean triple that one can ever imagine. One formula that can give us the triples is a = st, b = (s2 + t2)/2 and c = (s2 – t2)/2 (. A simple derivation of these formula will come from the main formula a2 + b2 = c2 (Silverman, 16 – 26). This is a shorten way to derive the formula from theorem 2.1(Pythagorean triples). a2 + b2 = c2  Ã‚  Ã‚  Ã‚  Ã‚   with a is odd, b is even and a, b and c have no common factors. a2 = c2 – b2   Ã‚  Ã‚  Ã‚   by additive property a2 = (c – b)(c + b)  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   by factoring (difference of two squares) by checking 32 = (5 – 4)(5 + 4) = 1*9 52 = (13 – 12)(13 + 12) = 1*25 72 = (25 – 24)(25 + 24) = 1*49

John Sayles Movie, Matewan :: Films Movies Essays

John Sayles' Movie, Matewan Writer and Director John Sayles movie Matewan, portrays the town of Matewan, West Virginia during their historic struggle out of feudalism in the 1920's. Because a single large company owned everything in Matewan, it was nearly impossible to rise out of the feudal relationship, especially since preaching the ideals of capitalism could easily result in losing all of one's property and material possessions. While feudalism can easily be written off as a economic and social structure of the past, this notion is easily proven false in Matewan; the people of Mingo County had no choice but to follow the orders of The Stone Mountain Coal Company. Hence, when the ideals of capitalism were introduced to the Matewan people by Joe Kinnehan they were stunned and shocked by what he was suggesting. Of course, The Stone Mountain Coal Company were more than shocked, they were outraged and adamant about not letting the workers of Matewan be part of any union. What do the feudal and capitalist economic systems represent? The Webster dictionary defines capitalism as, "an economic system characterized by open competition in a free market and by private or coperate ownership of the means of production and distribution." In Matewan, the idea of capitalism was first introduced by communist and union man himself, Joe Kinnehan. He urged the coal miners to realize the choices they could make and not to let a coal company make the decisions for them. He explained that this would entail all joining together and standing as one while demanding higher wages and better worker conditions. Not until the citizens of Matewan comprehend the feudal system they are living under, can they hope to change. The story of Matewan West Virginia portrays the struggle that an entire town underwent before they could release themselves from the grip of The Stone Mountain Coal Company. The coal miners of Matewan were bound in a feudal relationship and had no choice but to stay in it because The Stone Mountain Coal Company owned the entire town. Without continuing to work for the coal company, the residents of Matewan were sure to starve.

Ethics of Same Sex Marriage Essay

Society has many views on same sex relationships and marriages, people are either for it or against it. In this paper, we will first look at same sex marriage in America and how homosexuals and heterosexuals feel about the issue. We will look into the ethical issue that same sex marriage presents. We will look at how the classical theory of how deontology would resolve the issue of same sex marriage. Next, we will contrast deontology with the perspective of relativism. Finally, we will see which of these views on same sex marriage is closest to my own personal views. The ethical reason behind why people in society believe that same sex marriage is wrong is simply due to discrimination. Society does not have a valid reason why same sex marriage should not be allowed; it is just simply based on one’s own biased reasoning for not allowing it. Same sex marriage is something that people have been fighting for rights for many decades. Andrew Koppelman (2004), celebrated journalist and author, states that â€Å"Most Americans agree with the first sentence of the proposed amendment: â€Å"Marriage in the United States shall consist only of the union of a man and a woman. The main question this raises is whether this rule is important enough to enshrine in the Constitution† (p. 4). Many Americans can agree that the first sentence of the amendment is broad and that it invalidates domestic partnership laws that allow same sex couples the rights of marriage without the name (Koppelman, 2004). Marriage is not just a word but rather an institution, which the amendment makes impossible for same sex marriage to have the rights to that institution. In 1996, the defense of Marriage Act (DOMA) prevented the federal government from recognizing same sex marriages and allows individual states to have the power to define their own family laws. This allows each individual state to choose whether to ignore the first sentence of the amendment or agree with the first sentence of the amendment when deciding whether or not make same sex marriage legal in one’s own state. With this being said, a governor can choose to allow same sex marriage in the state they govern; however, when that governor’s time has lapsed and another person takes over, that person can decide to remove same sex marriage. All of same sex couples that were married in that state are now no longer considered married in the eyes of the law anymore. Also if a same sex couple gets married in their current state that they reside in where same sex marriage is considered legal; then the couple decide to move to another state that does not allow same sex marriage, they are then not considered married in the eyes of the law in the current state that they are now living in (Koppelman, 2004). In The Limits to Union: Same-Sex Marriage and the Politics of Civil Rights, Jonathan Goldberg-Hiller (2002) writes that gay rights activists have been fighting for same sex marriage rights since 1970. In 1990, one gay couple and two lesbian couples applied for a marriage licenses in Hawaii at the department of health, several people in the states made efforts to challenge the marriage laws and were denied. In 1991, the three couples hired a local civil rights attorney proceeded to sue the state circuit court for violations of their rights of privacy and equal protection. The judge ruled that the same sex couples did not enjoy the right to marry, which followed with the couples filing for an appeal in 1993 with the state supreme court. The state supreme court made the first national ruling that rejecting same sex marriage applications was unconstitutional gender discrimination, but didn’t show a state interest. The state at the time felt that not allowing same sex marriages did not uphold moral values and protect children and filed for a motion to reconsider; however, it was rejected in the Supreme Court (Goldberg-Hiller, 2002). Like Hawaii, many same sex couples have fought their state circuit court all the way up to their supreme court and have been fighting for many years to legalize marriage in their state and have been unsuccessful. In 2004, Connecticut, Iowa, New Hampshire, Vermont, Massachusetts, New York, the District of Columbia and two Native American tribal jurisdictions have legalized same sex marriages. As the years go on, more and more same sex couples will continue to fight for their right to marry who they choose and more states will eventually allow same sex couples their rights to marry whomever they choose (Koppelman, 2004). Those that are oppose same sex marriage focus their reasoning on religious beliefs. Same sex couples do not want to be treated as second class citizens, they do not focus on what religion states; they just want to be treated as equal as heterosexuals when it comes to their right. Same sex couples believe if one is to bar any class of people from marrying whomever they choose, it then deprives them of their social institution; that many feel defines the most meaningful part of life, to marry someone one loves. Same sex couples believe that their relationships are no different than that of a heterosexual marriage. Same sex couples can have maintain a home together, provide an environment that children can thrive in and care for each other the same as heterosexual married couples do (Goldberg-Hiller, 2002). In Attributions and the Regulation of Marriage: Considering the Parallels between Race and Homosexuality, Mark Joslyn and Donald Haider-Markel (2005) writes that for many people, these days, the issue of same sex marriage is an ethical controversy. Same sex relationships have been considered taboo and an ethical issue in many places throughout the United States. Many people oppose same sex marriage and the rights of homosexuals. The future for same sex marriage and civil unions appears to be very bright for legal recognition. Lesbians and gays had major setbacks in 2004 election, however, many feel that was just a speed bump. In Ethics and Social Responsibility, Kurt Mosser (2010) explains that ethics are the concern of what is morally right or wrong to an individual. Ethics is the study of what I ought to do or what should other people do. The philosophy known as ethics forces individuals to consider whether the things we do are right or wrong, good or bad, immoral or moral. Ethical issues have relationships even with religious traditions and legal political doctrines (p. 2). In Marriage, Autonomy, and the Feminine Protest, Debra Bergoffen (1999) explains that the ethical issues and the ethical problems same sex marriage presents is that society feels that if they make same sex marriage legal, the same sex couple would then destroy the meaning of marriage. To society, that meaning of marriage is procreation and the education of children. Procreation is not possible with the same sex, for it takes a man and a woman to make a child. Marriage is considered the ethical site of a couple and a decision people make to maintain a particular way of being. So, society feels that since same sex partners cannot procreate and have children that they should not be able to marry. Some men and women cannot conceive a child and procreate due to medical problems from either the man or woman. Many heterosexual partners, with the help of technology these days, use ways of fertility such as artificial insemination, egg donations, surrogate mothers and adoptions. So procreation is not always feasible with heterosexuals. So the statement of that marriage should be only among man and woman for the sake of procreation is actually unjust. A lesbian couple can then have a child with the help of sperm donation and a gay couple can have a child with the help of surrogacy or adoptions (Joslyn & Haider-Markel, 2005). If heterosexuals have the same issue as homosexuals, in regards to procreation, then one can say ethically it is fair to allow same sex partners to then marry To society, these days it is not uncommon to walk down the street and see a man and man, or woman and woman holding hands and it is considered normal to many Americans. However, to the older generations, they feel that a man and man, or woman and woman should not be together, even though it is a part of society these days; they have a hard time agreeing and understanding same sex relationships. The ethical values of someone from the sixties are going to be different from someone who was brought up in this day and age (Bergoffen, 1999). Ethics allows one to determine what is right and wrong, however, what we consider right or wrong is based on our upbringing and surrounding culture with other theories like deontology, we can see how these ways of life can be incorporated into society. The classical theories of deontology would resolve the problem of same sex marriage. Deontology looks at the reason and rule for why an act was done, instead of the consequences from the act. Deontology focuses on what we are obliged to do as moral human beings. Deontology realizes that all actions have consequences; however, those consequences whether or not actions are ethical should not be determined by the actions consequences. Deontologists feel that people have an obligation or duty to treat other human beings with respect, dignity and take their dignity into consideration when one has to deal with another person, as we expect them to do when someone has to deal with us. One cannot use another person nor can another person use them to get what one wants (Mosser, 2010). Deontological theory of how one should be treated allows same sex couples to be treated as just and fair as heterosexuals are treated. With deontology, people have the right to be who they are and is not fair to outcast others for any reason (Mosser, 2010). With this way of thinking, society should legalize same sex marriages and then all will be treated fairly and justly. When one says that a person cannot marry someone because they are marrying someone of the same sex, as oppose to opposite sex, is not treating someone with respect and dignity. Treating a person this way is instead out casting them as different and who are we to make that judgment in society? Deontology would fix this way of thinking in society, for people would treat same sex couples the same as heterosexual couples, thus allowing people to love and marry who they choose, not who society says they should marry. When one contrasts the theory of deontology with the perspective of relativism, one looks at two views that kind of work together. Where deontology focuses on what we are obliged to do as moral human beings, where relativism focuses on an individual’s moral claims that are either right in a culture or wrong for society. Relativism is where an individual’s values and beliefs are simply understood in one’s own culture, society or one’s own personal values. With relativism, one may find oneself debating with another person over what sport is considered the best sport. One person may believe that their viewpoint is more superior than the others view point of the issue. It is simply just that one person was raised with different views than the other. Deontology would then come in and would condemn some actions, if those actions violate the fundamental rule of treating others fairly and justly; thus allowing individuals to have their own beliefs or views (Mosser, 2010). To look at relativism and deontology together in regards to same sex marriages would help societies with dealing with same sex marriage. If society looked at the issue through the view of deontology, they would look at the situation in a moral way that every human being deserves to be treated fairly no matter the situation. Society with the perspective of relativism would have one’s own views on the matter of same sex relationships and marriage based on ones upbringing and culture and would accept what the culture says is ethical. So, a society that outlaws same-sex marriage would be acceptable to a relativist. However, with deontology in mind, society would treat same sex partners the same as heterosexuals. Even though each individual has one’s own views and beliefs on same sex marriage; with deontology one is not allowed to consider the consequences of same sex marriage, instead to simply treat others as human beings with the same respect and dignity that one would want done unto them (Mosser, 2010). My views on same sex marriage go along with deontology and relativism. I was raised in a family where we were taught certain ways of life but not to judge others for the way of life another may choose to live. With relativism, one would look at every situation based on how ones culture and ociety around them brought them up. For me, same sex marriage is something I feel should be allowed. I grew up having many gay and lesbian friends and to me it is normal for someone to date someone of the same sex. I do also feel that just because it is something that another person feels is right does not mean I have to agree with it to make it happen. When another person marries the same sex, it is them that are in that relationship not anyone else, so why does it offend or hurt others to see someone happy? Even if it is something a person is not comfortable with, who are they to judge? Would society be alright with someone telling them who they can or cannot marry? Some cultures marriage is pre-arranged; however, for many people in the United States heterosexuals are free to marry whomever they choose too. With deontology and relativism in mind, society can have their views on same sex marriage, but not let one’s own views cause them to treat others different than one would want to be treated. In my freshman year in high school, I moved from California to Virginia where I was introduced to a whole new way of life compared to ways of life I knew in California. With relativism, I was used to certain ways of being that to me made Virginia a strange place to live in. It was in high school that I encountered my first homosexual person. At first, the whole way of thinking to me was wrong, immoral and not how the bible said relationships should be. I now see that growing up, my mother thought in ways of deontology and would always explain to me, whenever I was confused, about ethical values and how whether we feel something is morally right or wrong it does not make others ways of life morally wrong. As the years past, I was one of the bridesmaid’s in a wedding of my two dear lesbian friends, held in Las Vegas. Even though their way of life was not the way I choose to live, it did not give me the right to judge them for who they loved and who they wanted to marry. At their wedding, I could see that the two were happy and in love and a year later, with the sperm donations of our friend, the two had a beautiful baby girl. With relativism and ethical upbringing, ones views on who they marry is their own views; with deontology, one accepts everyone for their own views and upbringing even if they do not understand it but one does not judge it. In conclusion, we have seen now that the ethical reason behind why people in society believe that same sex marriage is wrong is simply due to discrimination. We have, also, seen that society does base their belief of ame sex marriage on one’s own biased reasoning, not because it is harmful to society or anything that can be justified. It is simply a view of relativism, one’s own beliefs based on their society and upbringing. We have also seen that if society looked at same sex marriages with the views of deontology, society would be able to see things as they are and not discriminate and judge others for their beliefs and views. Society would treat everyone as they would want to be treated and same sex couples would be allowed to choose who they want to marry, just as heterosexual couples choose who they want to marry.

Descriptive Essay The Essay - 782 Words

Transience is the meaning of Japanese â€Å"sabi†. We often sense a beauty of time from antiques with scratched surface or muted colors. This kind of beauty is similar with â€Å"sabi† in that it is beyond appearance. Sabi is an aestheticism that embodies natural perspectives, and casts doubts on permanence. Therefore, we can say sabi means transient beauty like the fall and death of sakura, and also like the splendid fireworks. The view Sabi appreciates the washed-out beauty of time and the precious residue of time. 4.3.2 Examples The natural perspectives of Sabi find themselves in daily artifacts and pottery. The warmth and emotion of artisans integrates with their products through natural materials, hence making every piece a unique embodiment of different emotional attachment. The users of could also detect the inner emotion on touch of the products and a connection was created. The Japanese phrase â€Å"çˆ ±Ã§ â‚¬Ã¢â‚¬ Ã¯ ¼Ë†ai chakuï ¼â€°is an explanation on love or attachment created by artifacts, where â€Å"çˆ ±Ã¢â‚¬  means love and â€Å"ç â‚¬Ã¢â‚¬  means attachment, which also means the place where love is located. Chapter 4.4 Chapter Conclusion Till now, we have covered all aspects of Wabi-Sabi and their applications in real life. Rather than aesthetic, Wabi-Sabi is more like a design thinking. It is a different way to perceive our surroundings and a mean to treat them. Chapter 5 Thesis conclusion and induction of final project Chapter 5.1 Thesis conclusion As a Chinese student studying design in Pratt, I often found myselfShow MoreRelatedDescriptive Essay2288 Words   |  10 PagesHow to Write a Descriptive Essay More than many other types of essays, descriptive essays strive to create a deeply involved and vivid experience for the reader. Great descriptive essays achieve this affect not through facts and statistics but by using detailed observations and descriptions. What do you want to describe? As you get started on your descriptive essay, its important for you to identify exactly what you want to describe. Often, a descriptive essay will focus on portraying one of theRead Moredescriptive essay799 Words   |  4 Pagesï » ¿Fred Cotten Descriptive Essay September 27, 2014 English 1010 - 85 A Mini Vacation to Atlanta, Georgia Traveling is one of my family’s favorite things to do. The family has visited numerous places throughout the United States, however, none are as memorable as Atlanta, Georgia. In Atlanta, there are many places to go and sights to see such as: Cola-cola factory, Cabbage Patch Kids Factory, Under Ground Mall, the Zoo, Atlanta Braves Stadium, Six Flags Over Georgia, Stone Mountain Park, andRead MoreDescriptive Essay971 Words   |  4 PagesChandell Gabler English 099-22-Intro to College Writing 9:05-10:15 am Professor Braxton-Robinson/Professor Sheffield Assignment: Write a Descriptive Essay describing a storm you have witnessed. The storm I witnessed was Hurricane Sandy. When I first heard of the hurricane I thought it wouldnt be as bad as some people were prediciting, but as the storm approached I certainly changed my mind. As we sat in the house it became very real how bad the storm was going to be. It was scary toRead MoreDescriptive Essay865 Words   |  4 PagesDescriptive Essay A Horrid Memory My sweaty palms ran through-out the thin strands of my hair as I tried to comfort myself for just a single moment. I felt scared, almost terrified as the screams of my nightmare echoed in my ears and made my head throb. Almost never did I have such vivid dreams, and when I did, I never remembered them the next day. Nevertheless, I dont think that Ill ever be able to make amends with my darkest memories and reliving situations that scarred my childhoodRead MoreDescriptive Essay652 Words   |  3 PagesDescriptive Essay It was a seemingly beautiful day as I drove on a joy ride in my bright blue sports car, with the wind in my hair. This is an everyday activity for me. My friends would mock me saying, â€Å"You spend more money on gas than you make working.† I laughed to myself at the thought of jokes amongst my friends. That happiness was soon halted. As I continued driving, I saw a car coming rapidly around the curve. The car did not show any signs of slowing down as it headed towards the intersectionRead MoreDescriptive Essay787 Words   |  4 PagesDescriptive Essay September 27, 2014 English 1010 - 85 A Mini Vacation to Atlanta, Georgia Traveling is one of my family’s favorite things to do. The family has visited numerous places throughout the United States, however, none are as memorable as Atlanta, Georgia. In Atlanta, there are many places to go and sights to see such as: Cola-cola factory, Cabbage Patch Kids Factory, Under Ground Mall, the Zoo, Atlanta Braves Stadium, Six Flags Over Georgia, Stone Mountain Park, and the Atlanta AquariumRead MoreDescriptive Narrative Descriptive Essay1093 Words   |  5 Pages Title The alluring sun rays peak through the clouds at Ponder Park as I walk on the damp grass. The sweet faces of those who have come peer off into the distance. This heinous, yet endearing place I have arrived at blows a cool breeze through-out my entire body. Tiring, however its small glimpses of beauty glorifies me. As I begin to stretch my legs to run I feel the tightness of my thigh begin to loosen. I hear the mumbles of all of my teammates as they surround near me. The wind playsRead MoreDescriptive Narrative Descriptive Essay964 Words   |  4 PagesIt had been like dying, that sliding down the mountain pass. It had been like the death of someone, irrational, that sliding down the mountain pass and into the region of dread. It was like slipping into fever, or falling down that hole in sleep from which you wake yourself whimpering. We had crossed the mountains that day, and now we were in a strange place – a hotel in central Washington, in a town near Yakima. The ecli pse we had traveled here to see would occur early in the next morning. I layRead MoreNarrative and Descriptive Essay1226 Words   |  5 Pages Compare-Contrast Essay Eng121: English Composition I (AXC13480) Regina McKinney Professor: Nancy Segovia January 1, 2014 A narrative essay is about storytelling for a narrative story to work it must capture and hold the audience attention you must give a clear understanding of your story. A descriptive essay lets you describe in detail what the essay is all about using words that appeal to your sense of smell, hearing, see, touch, and taste. A descriptive essay lets you use words thatRead MoreDescriptive Essay2920 Words   |  12 PagesJennifer Schacht ENG-090 2/10/2011 Descriptive Essay Final We have been waiting nine long months, and we have had much preparation to do before the arrival of our daughter. Of all the things we have prepared for her, I am most proud of her room, my mother and I painted it and sat it up with all the accessories together. I sat on the floor and taped off the stripes to be painted, and my mom painted. It was a great team effort, and we are so happy how it turned out. With flowers and butterflies